New edge does not split face
By selecting the vertices and pressing F you are simply creating a new edge. The faces connected to the vertices are not changed, this behaviour correct.

In this position, delete the problematic face.Select all the vertices and/or edges, which need to be fill and press 
F to fill them with new faces.

Alternatively, before creating the edge which intersects the face, press K to cut the face with the Knife Tool. The knife tool can snap to the vertices along the faces' edge.